Student's T-test - Wikipedia

 

 

One-sample t-test

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A one-sample Student's t-test is a location test of whether the mean of a population has a value specified in a null hypothesis. In testing the null hypothesis that the population mean is equal to a specified value μ0, one uses the statistic

t = x ¯ − μ 0 s / n , {\displaystyle t={\frac {{\bar {x}}-\mu _{0}}{s/{\sqrt {n}}}},}  

where x ¯ {\displaystyle {\bar {x}}}   is the sample mean, s is the sample standard deviation and n is the sample size. The degrees of freedom used in this test are n − 1. Although the parent population does not need to be normally distributed, the distribution of the population of sample means x ¯ {\displaystyle {\bar {x}}}   is assumed to be normal.

By the central limit theorem, if the observations are independent and the second moment exists, then t {\displaystyle t}   will be approximately normal N ( 0 , 1 ) {\textstyle {\mathcal {N}}(0,1)}  . This is only an approximation as the central limit theorem would apply to t if s was the actual standard deviation of x, while it is the sample standard deviation as the actual standard deviation is not generally known. Therefore, t asymptotically follows a Student's t-distribution.

Two-sample t-tests

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A two-sample location test of the null hypothesis that the means of two populations are equal. All such tests are usually called Student's t-tests, though strictly speaking that name should only be used if the variances of the two populations are also assumed to be equal; the form of the test used when this assumption is dropped is sometimes called Welch's t-test. These tests are often referred to as unpaired or independent samples t-tests, as they are typically applied when the statistical units underlying the two samples being compared are non-overlapping.[14]

Two-sample t-tests for a difference in means involve independent samples (unpaired samples) or paired samples. Paired t-tests are a form of blocking, and have greater power (probability of avoiding a type II error, also known as a false negative) than unpaired tests when the paired units are similar with respect to "noise factors" (see confounder) that are independent of membership in the two groups being compared.[15] In a different context, paired t-tests can be used to reduce the effects of confounding factors in an observational study.

Independent (unpaired) samples

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The independent samples t-test is used when two separate sets of independent and identically distributed samples are obtained, and one variable from each of the two populations is compared. For example, suppose we are evaluating the effect of a medical treatment, and we enroll 100 subjects into our study, then randomly assign 50 subjects to the treatment group and 50 subjects to the control group. In this case, we have two independent samples and would use the unpaired form of the t-test.

Paired samples

edit Main article: Paired difference test

Paired samples t-tests typically consist of a sample of matched pairs of similar units, or one group of units that has been tested twice (a "repeated measures" t-test).

A typical example of the repeated measures t-test would be where subjects are examined prior to a treatment, say for high blood pressure, and the same subjects are examined again after treatment with a blood-pressure-lowering medication. By comparing the same patient's numbers before and after treatment, we are effectively using each patient as their own control. That way the correct rejection of the null hypothesis (here: of no difference made by the treatment) can become much more likely, with statistical power increasing simply because the random inter-patient variation has now been eliminated. However, an increase of statistical power comes at a price: more examinations are required, each subject having to be examined twice.

Because half of the sample now depends on the other half, the paired version of Student's t-test has only ⁠n/2⁠ − 1 degrees of freedom (with n being the total number of observations). Pairs become individual test units, and the sample has to be doubled to achieve the same number of degrees of freedom. Normally, there are n − 1 degrees of freedom (with n being the total number of observations).[16]

A paired samples t-test based on a "matched-pairs sample" results from an unpaired sample that is subsequently used to form a paired sample, by using additional variables that were measured along with the variable of interest.[17] The matching is carried out by identifying pairs of values consisting of one observation from each of the two samples, where the pair is similar in terms of other measured variables. This approach is sometimes used in observational studies to reduce or eliminate the effects of confounding factors.

Paired samples t-tests are often referred to as "dependent samples t-tests".